A254526 Fourier expansion of first basis element of space of weight 1 modular forms on Gamma1(12).

1, 0, 0, 0, 0, 12, 0, -12, 0, 0, 12, 0, 0, 0, -12, 12, 0, 12, 0, -12, 12, -12, 0, 0, 0, 12, 0, 0, -12, 12, 12, -12, 0, 0, 12, 0, 0, 0, -12, 0, 12, 12, -12, -12, 0, 12, 0, 0, 0, -12, 12, 12, 0, 12, 0, 0, -12, -12, 12, 0, 12, 0, -12, -12, 0, 24, 0, -12, 12, 0

Offset

0,6

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of 3 * (phi(q)^2 + phi(q^3)^2) / 2 - (a(q) + a(q^2)) in powers of q where phi() is a Ramanujan theta function and a() is a cubic AGM theta function.

Moebius transform is period 12 sequence [ 0, 0, 0, 0, 12, 0, -12, 0, 0, 0, 0, 0, ...]. - Michael Somos, Jan 31 2015

Example

G.f. = 1 + 12*q^5 - 12*q^7 + 12*q^10 - 12*q^14 + 12*q^15 + 12*q^17 - 12*q^19 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[n==0], 12 Sum[ Boole[ Mod[d, 12] == 5] - Boole[ Mod[d, 12] == 7], {d, Divisors @ n}]];

Prog

(PARI) {a(n) = if( n<1, n==0, 12 * sumdiv( n, d, (d%12 == 5) - (d%12 == 7)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 3 * eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^6 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^12 + A)) - 2 * eta(x^2 + A)^6 * eta(x^3 + A) / (eta(x + A)^3 * eta(x^6 + A)^2), n))};
(Magma) Basis( ModularForms( Gamma1(12), 1), 70)[1];

Crossrefs

Sequence in context: A370526 A048730 A307163 * A156390 A059680 A307170

Adjacent sequences: A254523 A254524 A254525 * A254527 A254528 A254529

Keyword

sign

Author

Michael Somos, Jan 31 2015

Status

approved